An Operational Approach to Quantum Probability

Abstract

In the two preceding papers ‘Completeness of Quantum Logic’ (CQL) and ‘Quantum Logical Calculi and Lattice Structures’ (QLC) an operational approach to formal quantum logic was developed. Beginning with a pragmatic definition of quantum mechanical propositions by means of material dialogs a formal dialog-game was introduced for establishing formally true propositions. It was shown in CQL that the formal dialog-game can be replaced by a calculus Teff of effective (intuitionistic) quantum logic which is complete and consistent with respect to the dialogic procedure. In QLC we showed that Teff is equivalent to a propositional calculus Qeff Since the calculus Qeff is a model for a certain lattice structure, called quasi-implicative lattice (Leff), the connection between quantum logic and the quantum theoretical formalism is provided. Lqi is a weaker algebraic structure than the orthomodular lattice of the subspaces of a Hilbert space Lq which can be interpreted as the pro-positional calculus of value-definite quantum logic. This establishes a quantum logical interpretation of Lq.